What is Arithmetic Mean in Statistics?

Statistics is a fascinating and vital topic to study. It entails a thorough examination of facts presented in the form of numbers. Statistics assist us in analyzing and developing conclusions from a data collection. The calculation of numerous arithmetic values is a part of statistics. Geometric mean, arithmetic mean, mode median, and many more values can be used to analyze data. In statistics, the arithmetic mean can be calculated for any given set of data, regardless of its size. The collection is typically a set of results from an experiment or observational research, or a group of survey results. In some areas of mathematics and statistics, the name “arithmetic mean” is favored since it distinguishes it from other means such as the geometric and harmonic means.

Arithmetic Mean

The arithmetic mean is a number calculated by dividing the total of a set’s elements by the number of values in the set. So you can use the phrase “average” or “arithmetic mean” if you want to be more formal. To determine a central value for a group of values, arithmetic employs two basic mathematical operations: addition and division. A bar (x̄) is commonly used to represent it. 

Arithmetic Range

The difference between the highest and smallest value of data is represented by the term range. This allows us to determine the data’s distribution range.

Arithmetic Mean Formula for Ungrouped Data

Sum of observations/Number of observations = Arithmetic mean

Range Formula

Highest value – Lowest value = Arithmetic Range

Arithmetic Mean for Grouped Data

The arithmetic mean for grouped data can be calculated using three approaches (Direct method, Short-cut method, and Step-deviation method). The approach to be utilized is determined by the numerical values of xi and fi. The sum of all data inputs is xi, and the frequency sum is fi. The sign Σ(sigma) stands for summation. The direct method will work if xi and fi are small enough. We employ the assumed arithmetic mean approach or step-deviation method if they are numerically large.

Direct Method

Let x1, x2, x3,……xn be the observations with frequency f1, f2, f3,……fn. The mean is then determined using the following formula:

x = (x1f1+x2f2+……+xnfn) /Σfi

The total of all frequencies is represented by f1+ f2 +….fn = Σfi.

Short-Cut Method

The assumed mean method, also known as the change of origin method, is a short-cut approach. This procedure is described in the stages below.

Step 1: Determine the class marks (middle) for each class (xi).

Step 2: Let A signify the data’s assumed mean.

Step 3: Calculate the deviation (di) = xi – A.

Step 4: Apply the following formula:

X̄ = A + ( Σfidi / Σfi )

Step-Deviation Method

This is also known as the scale method or change of origin method. This procedure is described in the stages below:

Step 1: Determine each class’s grade point average (xi).

Step 2: Let A signify the data’s assumed mean.

Step 3: Calculate ui = (xiA)/h, where h is the number of students in the class.

Step 4: Apply the following formula:

X̄ = A + h * ( Σfiui / Σfi )

Advantages of Arithmetic Mean

• Because the formula for calculating the arithmetic mean is fixed, the result remains constant. It is unaffected by the position of the value in the data set, unlike the median.

• It takes into account each value in the data set.

• Finding an arithmetic mean is quite simple; even a layperson with no finance or math skills can do it.

• It’s also a good measure of central tendency because it produces useful results even with large groups of numbers.

• Unlike mode and median, it can be subjected to a variety of algebraic treatments. The mean of two or more series, for example, can be calculated from the mean of the individual series.

• The arithmetic mean is also widely used in geometry. The arithmetic mean of the coordinates of the vertices, for example, is the coordinates of the “centroid” of a triangle (or any other figure bounded by line segments).

Limitations of Arithmetic Mean

• The most significant disadvantage of the arithmetic mean is that it is influenced by the data set’s extreme values.

• The value of the mean in a distribution with open-end classes cannot be determined without making assumptions about the size of the class.

• It’s nearly impossible to find the arithmetic mean visually or graphically.

• It can’t be used to collect qualitative data like honesty, favorite milkshake taste, most popular product, and so on.

• If a single observation is missing or lost, we can’t find the arithmetic mean.

Conclusion

The arithmetic mean is a number calculated by dividing the total of a set’s elements by the number of values in the set. So you can use the common term “average” or the more formal term “arithmetic mean.” It’s your choice; they both signify the same thing. The difference between the highest and smallest value of data is represented by the term range. Using the preceding example as an example, this allows us to calculate the range over which the data is scattered.